## Fecundity and Fertility in Population Dynamical Models

Being a cornerstone of population biology, quantifying fecundity and fertility patterns is a central component of models that attempt to describe complex population dynamics and structure. Many models exist exclusively to provide direction in the understanding of the evolution of life-history strategies, while others have a more utilitarian function such as the estimation of a population's future status for conservation purposes. One of the most common methods used to summarize population behavior mathematically is the stage-structured matrix population model that provides a link between the individual (and its selective pressures) and the population using stage-specific estimates of vital rates (rates of birth, growth, maturation, fertility, and mortality). A more detailed description of matrix models is provided in this encyclopedia; however, there are some important aspects of these models that deal with fecundity and fertility specifically.

Stage-structured models provide an estimate of the stable stage distribution, which is the theoretical proportional allocation of individuals within the population to the defined stages (e.g., age groups, size classes, developmental stages) resulting from constant demographic rates. The stable stage distribution is a useful metric because it provides the theoretical composition of a population exhibiting a fixed birth rate, so factors such as environmental variation or intrinsic regulation that alter this theoretical distribution can be ranked for their effects on the future composition of a population. This introduces the concept of'reproductive value' - a measure of the combined effects of fecundity, fertility, and survival that takes an individual's proportional contribution to the future status of its population into account. It is formally expressed as the sum of the current and future reproductive values, and is considered the currency used by natural selection to produce a particular life-history strategy.

Because life histories evolve to maximize reproductive success, fecundity is linked formally to population models by using the measure of reproductive value. In matrix models the relative effects of proportional changes in fertility (and survival) to population growth provide a sensitivity analysis for stage-specific terms. Here, the reproductive value for a particular stage is calculated as the product of the sensitivity of all matrix elements that contain that stage and the stable stage proportion. For example, short-lived species tend to demonstrate a relatively higher sensitivity for fertility than survival, while the reverse is often true for long-lived organisms. Thus, sensitivity analysis can be used in combination with observed variation in demographic rates to determine which factors have the largest impact on population growth and thus, the future composition of the population. Lifetime reproduction patterns expressed as reproductive allocation and ranked relative to other demographic rates therefore provide a means to compare the evolution of life-history strategies.

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